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Revision difference : parallelism of two planes |
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| Two planes $\pi$ and $\varrho$ in the 3-dimensional Euclidean space are {\em parallel}\, iff they either have no common points or coincide, i.e. iff |
Two planes $\pi$ and $\varrho$ in the 3-dimensional Euclidean space are {\em parallel}\, iff they either have no common points or coincide, i.e. iff |
| $$\pi\cap\varrho \;=\; \varnothing \quad \mbox{or} \quad \pi\cap\varrho\;=\; \pi.$$ |
$$\pi\cap\varrho \;=\; \varnothing \quad \mbox{or} \quad \pi\cap\varrho\;=\; \pi.$$ |
| An \PMlinkname{equivalent}{Equivalent3} condition of the parallelism is that the normal vectors of $\pi$ and $\varrho$. are parallel. |
An \PMlinkname{equivalent}{Equivalent3} condition of the parallelism is that the normal vectors of $\pi$ and $\varrho$. are parallel. |
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If the planes have the equations \,$A_1x+B_1y+C_1z+D_1 = 0$\, and\, $A_2x+B_2y+C_2z+D_2 = 0$,\, the parallelism means the \PMlinkname{proportionality}{Variation} of the coefficients of the variables:\, there exists a \PMlinkescapetext{constant} $k$ such that
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If the planes have the equations \,$A_1x+B_1y+C_1z+D_1 = 0$\, and\, $A_2x+B_2y+C_2z+D_2 = 0$,\, the parallelism means the \PMlinkname{proportionality}{Variation} of the coefficientss:\, there exists a \PMlinkescapetext{constant} $k$ such that
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$$A_1 = kA_2, \quad B_1 = kB_2, \quad C_1 = kC_2_2.$$
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$$A_1 = kA_2, \quad B_1 = kB_2, \quad C_1 = kC_2, \quad D_1 = kD_2.$$
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